\chapter{Mathematical Background}

\section{Information entropy}

entropy

\section{Equivalence relation}

See \url{http://en.wikipedia.org/wiki/Equivalence\_relation}.

\section{Permutation}

[permutation notations, properties]

[equivalence relation, equivalence class, etc.]

A \emph{permutation} is a bijection (one-to-one mapping) of a set onto itself.\footnote{
More details can be found at \url{http://en.wikipedia.org/wiki/Permutation}.}
For example, consider a set containing six elements. You can think of them as the six colors in a Mastermind game. For convenience, label each element with an index starting from one. A possible permutation is the following:
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
5 & 2 & 1 & 6 & 3 & 4
\end{pmatrix} ,
\]
where the first row displays the elements in the set, and the second row displays the \emph{images} of the elements under the permutation, i.e.\ the element that each element is mapped to.

The above notation for the permutation can be abbreviated into one line by only keeping the second row, which becomes $(5 2 1 6 3 4)$.

A permutation can be decomposed into a \emph{product} of disjoint \emph{cycles}, which partitions the elements of the set into parts where the elements in each part can be permuted independently and then combined to form the complete mapping. For example, the above permutation can be decomposed into the product of three cycles:
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
5 & 2 & 1 & 6 & 3 & 4
\end{pmatrix} 
= (1 5 3) (2) (4 6) .
\]
It is easy to see that the cycles can be commuted and the elements in a cycle can be rotated without changing the overall permutation. Up to these differences, such decomposition is unique.

A permutation can be inverted. To find the inverse of a permutation, simply exchange the two roles in the notation and then sort the upper row. In the above example, the inverse permutation is
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
3 & 2 & 5 & 6 & 1 & 4
\end{pmatrix} .
\]

%Two permutations of the same size can be compounded to form a composite permutation. Let @c P be the composite of permutation
% * <code>P<sub>1</sub></code> and <code>P<sub>2</sub></code>.
% * The effect of applying @c P is equivalent to first applying
% * <code>P<sub>1</sub></code> followed by applying <code>P<sub>2</sub></code>.
% * The notation to write a composite permutation can be confusing,
% * so we omit it here.

A \emph{partial permutation} on a set is a bijection between two subsets of it.\footnote{For more details, see \url{http://www.maths.qmul.ac.uk/~pjc/odds/partial.pdf}.}
For example, a partial permutation of the above (complete) permutation could be
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
* & 2 & * & * & 3 & 4
\end{pmatrix} ,
\]
where the asterisks denote unmapped elements, sometimes known as ``holes'' of the permutation. 

It is easy to see that any partial permutation can be \emph{extended} to form a complete permutation. However such extension is not unique. For example, there are $3! = 6$ ways to extend the above partial permutation, one of which that differs from the original example could be
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
1 & 2 & 5 & 6 & 3 & 4
\end{pmatrix} .
\]

\section{Graph isomorphism}

\chapter{Listing of Selected Strategies}

Below we list several selected strategies for the standard Mastermind game (4 pegs, 6 colors, repetition allowed). These strategies appear in the journal articles of the respective authors. The format is the conventional Irving format.\footnote{Several typos in the original articles have been corrected. In case where the original format is different, it is converted to Irving's format.}

\section{Comparison of Strategies}

display a set diff of the strategy trees

\section{Selected Strategies for Mastermind}

\subsection{Knuth (1976)}

This strategy appears in \cite{knuth76}. It is a heuristic strategy that applies the min-max heuristic function. [more precise details needed] The strategy tree is listed below.

%\begin{lstlisting}{breaklines=true}

%\begin{quote}
{ \scriptsize
1296(1122: 1, 16(1213: 0, 0, 0, 0, 0; 1, 4(1415), 3(1145), 0; 1, 3(4115), 3(1145); 0, 1; 0), 96A, 256B, 256C; 0, 36D, 208E, 256F; 4(1213), 32G, 114H; 0, 20I; 1)

A = (2344: 0, 2, 16 (3215: 0, 0, 0, 0, 0; 1, 2, 1, 1; 2, 3(3231), 2; 0, 3(3213); 1), 
14(5215: 0, 0, 0, 0, 0; 0, 1, 3(3511), 3(3611); 1, 1, 2; 0, 2; 1), 4(1515); 0, 6(2413), 
18(2415: 1, 1, 0, 0, 0; 1, 2, 3(2253), 3(2236); 1, 2, 2; 0, 1; 1), 15(2256x); 0, 4(2234), 14(3315x); 0, 3(2314); 0) 
}
%\end{quote}

B = (2344: 0,7(2335),41(3235: 0,0,2,3(4613),2;0,3(5263),6(3413),6(3416);2,4(3256),6(1336);0,6(1536);1), 
    44(3516: 1,4(4651),6(6255),1,0;3(5613),7(1461),5(4551),1;3(1113),5(3551),3(4515);0,4(1145);1), 
    16(5515: 0,0,1,1,0;0,2,2,1;1,1,3(1516);0,3(1516);1);
    2,21(3245: 1,3(2436),0,0,0;2,2,2,0;2,3(3234),2;0,3(3243);1), 
    42(4514: 1,1,7(2456),4(2635),3(2636);0,4(1356),5(4361), 6(1635) ;2,2,3(3614);0,3(4414) ;1), 
    34(3315: 0,0,3(5641),4(2566),1;1,4(5361),4(5614),5(6614);2,4(3331),1;0,4(3316) ;1); 
    3(2434),13(2425x),23(1545: 0,1,3(2654),3(2353),4(1136) ;0,2,4(2564),3(2335) ;0,0,2;0,1;0); 
    0,9(1335x) ;1) 
C = (3345: 2,20(4653: 2,2,0,0,0;3(4536),3(4534),1,0;2,2,1;0,3(4453);1), 
    42(6634: 0,3(4566),4(4556),1,0;2,5(4656),6(5653),4(1444) ;2,5(5636),5(4654);0,4(1413) ;1), 
    16(6646: 0,0,1,0,0;0,3(1416),1,1;3(1416),3(5666),2;0,2;0),1; 
    4(3453),40(3454: 1,5(4535),6(1436),0,0;2,5(4356),6(3536),0;1,3(3564),6(3463);0,4(3456);1), 
    46(3636: 1,1,3(4364),6(4565),6(4544) ;0,5(4366), 6(1565),6(4546) ;2,4(3466),3(3556) ;0,2;1), 
    18(3656: 0,1,1,1,1;0,3(5665),3(6446),3(4446);0,1,3(4646) ;0,1;0); 
    5(3435x),20(3443: 0,0,4(4355),0,0;0,3(3334),4(3356),0;1,2,4(3455);0,1;1), 
    29(3636: 0,1,3(5365),4(6445),4(1444); 0,2,3(3565),4(4645) ;1, 1,4(3446);0,2;0) ;0,12(3446x);1) 
D = (1213: 1,4(1145),3(1415),0,0;0,6(1114x),7(2412x),0;2,4(1145),4(1145x);0,4(1114x);1)
E = (1134: 0,4(1312),24(3521: 1,2,4(4612),0,0;0,3(3312),3(2423),0;2,2,3(4621);0,3(3321);1), 
    38(2352: 2,4(3226),4(5621),1,0;1,5(2223),7(6242),1;2,4(2323),4(2462);0,2;1), 
    20(2525: 1,2,1,0,0;0,3(2252),3(2262),0;2,2,2;0,3(2225);1); 
    4(1341),34(1315: 1,3(4151),4(4161),0,0;1,6(6451),6(1461),0;3(1351),3(1361),2;0,4(1113);1), 
    32(1516: 2,2,3(2145),0,4(2324);2,4(1661),4(1245),0;3(1561),3(1551),1;0,3(1511);1), 
    22(1256: 1,0,4(2524),2,0;0,2,4(5224),4(2224);2,0,0;0,2;1);4(1314),12(1315x),12(1235x);0,2;0)
F = (1344: 0,7(1335),41(3135: 0,0,2,3(4623),2;0,3(5163),6(3423),6(3426);2,4(3156),6(1436);0,6(1536);1),
    44(3526: 1,4(4652),6(6155),1,0;3(5623),7(1462),5(4552),1;3(1123),5(3552),3(4525) ;0,4(1145) ;1), 
    16(5525: 0,0,1,1,0;0,2,2,1;1,1,3(1516);0,3(1516);1); 
    2,21(3145: 1,3(1436),0,0,0;2,2,2,0;2,3(3134),2;0,3(3143) ;1), 
    42(4524: 1,1,7(1456),4(1635),3(1636) ;0,4(1356), 5(4362),6(1336) ;2,2,3(3624) ;0,3(4424) ;1), 
    34(3325: 0,0,3(5642),4(1566),1;1,4(5362),4(5624),5(6624);2,4(3332),1;0,4(3326);1); 
    3(1434),13(1415x),23(1415: 0,0,2,4(3324),0;0,4(1546),4(1356),4(1136) ;0,2,3(1136) ;0,0;0) ;0,9(1335x) ;1) 
G = (1223: 1,4(2145),3(4115),0,0;0,5(2145),6(4512),0;2,4(1245),3(1415);0,3(1145);1) 
H = (1234: 2,16(1325: 1,3(4152),3(4162),0,0;1,3(3126),2,0;1,1,1;0,0;0), 
    20(1325: 0,3(5162),1,0,0;0,2,4(4522),4(4622);0,3(5125),3(2116);0,0;0), 
    6(2515),0;4(1323),21(1352: 0,1,2,0,0;2,4(1623),2,0;1,3(1323),3(1462);0,2;1), 
    16(2156x),12(1315x) ;2, 6(3526),8(1536x) ;0, 1;0) 
I = (1223: 0,0,0,0,0;1,5(1145x),4(1114x),0;1,3(1415),4(1114x);0,2;0) 
%\end{lstlisting}

\subsection{Irving (1979)}

\subsection{Neuwirth (1981)}

\subsection{Koyama (1993)}

\subsection{My Strategy}

\section{Selected Strategies for Bulls and Cows}

\subsection{Optimal Strategy}

Below is the result of a sample run of my program.

{ \scriptsize
5040 (1234: 9A, 264B, 1260C, 1440D, 360E; 8F, 216G, 720H, 480I; 6J, 72K, 180L; 24M; 1)

A = (2341: 2 (4123), 0, 0, 0, 0; 4 (4312: 1, 0, 0, 0, 0; 2 (2413), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 2 (2143), 0, 0; 0; 1)

B = (2546: 0, 18 (3452: 1, 2 (4623), 1, 0, 0; 1, 3 (6423), 3 (4162), 0; 2 (3425), 2 (4152), 1; 1; 1), 70 (4782: 0, 6 (3428: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (8421), 1; 1; 1), 6 (0423: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (0421); 1), 11 (3451: 0, 1, 1, 0, 0; 1, 3 (5321), 1, 0; 1, 0, 1; 1; 1), 0; 0, 9 (4821: 1, 1, 1, 0, 0; 0, 0, 0, 0; 1, 2 (4127), 1; 1; 1), 12 (4129: 1, 2 (3492), 1, 0, 0; 0, 0, 0, 0; 1, 2 (4021), 2 (4023); 2 (4120); 1), 11 (3162: 0, 1, 2 (4351), 0, 0; 1, 2 (5312), 1, 0; 1, 1, 0; 1; 1); 0, 6 (4172), 6 (4102: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (4192); 1); 3 (4382); 0), 56 (3728: 0, 2 (7312), 10 (4381: 1, 1, 0, 0, 0; 1, 1, 2 (9312), 0; 1, 1, 0; 1; 1), 10 (4391: 1, 1, 0, 0, 0; 1, 2 (4103), 0, 0; 2 (4193), 1, 0; 1; 1), 0; 0, 7 (3172), 13 (3192: 1, 1, 2 (4713), 0, 0; 1, 1, 3 (3481), 0; 1, 1, 0; 1; 1), 4 (3401); 0, 3 (3712), 5 (3129); 2 (3721); 0), 0; 0, 16 (4326: 2 (2463), 3 (6142), 3 (2451), 0, 0; 1, 1, 1, 0; 1, 1, 1; 1; 1), 42 (3571: 0, 3 (5143), 7 (6143), 12 (8149: 0, 1, 3 (2483), 1, 0; 0, 1, 1, 1; 0, 1, 2 (0142); 1; 0), 0; 0, 4 (5341), 6 (6341), 6 (1892*); 0, 1, 2 (2471); 1; 0), 32 (2178: 0, 0, 4 (8341), 4 (0341), 0; 0, 4 (2813), 9 (2013: 0, 0, 1, 0, 0; 1, 1, 2 (8143), 0; 1, 1, 0; 1; 1), 4 (3149); 0, 3 (2183), 3 (2193); 1; 0); 0, 6 (2345), 20 (2178: 0, 0, 0, 1, 0; 0, 2 (2841), 7 (1763: 0, 1, 1, 2 (2941), 0; 0, 0, 1, 1; 0, 0, 1; 0; 0), 5 (2340); 0, 1, 3 (2140); 1; 0); 4 (2541); 0)

C = (2356: 4 (6523: 0, 0, 0, 0, 0; 2 (3562), 0, 0, 0; 1, 0, 0; 0; 1), 64 (3782: 0, 6 (5823: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (7523), 1; 1; 1), 8 (5923: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (9623), 2 (6023); 2 (5023); 1), 12 (5463: 0, 1, 2 (6125), 0, 0; 1, 2 (6425), 1, 0; 2 (4563), 1, 0; 1; 1), 0; 0, 7 (3825: 0, 0, 0, 0, 0; 0, 1, 1, 0; 1, 2 (3527), 1; 0; 1), 6 (3025: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (3529), 1; 1; 1), 12 (3561: 0, 1, 1, 0, 0; 1, 3 (6512), 2 (5462), 0; 1, 1, 1; 0; 1); 0, 4 (3725), 6 (3692: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (3592); 1); 3 (3682); 0), 284 (6742: 1, 10 (7438: 0, 1, 1, 2 (4620), 0; 0, 1, 2 (5427), 1; 0, 1, 1; 0; 0), 48 (3468: 1, 2 (4603), 6 (1263*), 8 (4025: 0, 0, 0, 0, 0; 0, 0, 1, 2 (7923); 1, 1, 1; 1; 1), 3 (7521); 1, 2 (4963), 9 (3471: 0, 1, 1, 1, 0; 0, 1, 2 (4561), 1; 0, 1, 0; 1; 0), 6 (5429); 1, 2 (0463), 4 (5428); 2 (3469); 1), 56 (3168: 1, 2 (9613), 9 (3829: 0, 0, 2 (5483), 1, 0; 1, 1, 1, 0; 1, 1, 1; 0; 0), 10 (5021: 0, 0, 1, 1, 0; 0, 0, 2 (0923), 1; 1, 1, 1; 1; 1), 0; 1, 4 (3610), 7 (5173: 1, 0, 1, 2 (3829), 0; 1, 0, 1, 0; 0, 0, 0; 0; 1), 8 (5120); 3 (8163), 4 (3061), 4 (3028); 2 (3160); 1), 21 (3185: 2 (8513), 4 (5013), 0, 0, 0; 1, 6 (3501), 0, 0; 3 (5183), 2 (3915), 0; 2 (3105); 1); 3 (6427), 22 (4682: 1, 2 (6420), 2 (6127), 0, 0; 1, 2 (0462), 6 (4763), 0; 1, 2 (4962), 2 (7612); 2 (4602); 1), 45 (8162: 1, 2 (6921), 6 (3648: 1, 0, 1, 0, 1; 0, 0, 0, 0; 0, 0, 2 (5641); 0; 1), 10 (6493), 2 (7543); 1, 4 (0612), 8 (4872*), 6 (5402); 1, 0, 1; 2 (9162); 1), 35 (3845: 0, 0, 4 (6183), 10 (0512), 0; 1, 4 (0543), 4 (3982), 2 (3902); 2 (3548), 2 (3540), 3 (3892); 2 (3945); 1); 3 (4762: 0, 0, 0, 0, 0; 2 (7642), 0, 0, 0; 0, 0, 0; 0; 1), 9 (8642: 0, 0, 1, 0, 0; 1, 2 (6492), 1, 0; 0, 0, 1; 2 (0642); 1), 24 (2548: 0, 0, 4 (6182), 6 (6912), 1; 1, 2 (5042), 4 (5743), 2 (6043); 1, 2 (0542), 1; 0; 0); 6 (1728: 0, 0, 1, 2 (6042), 0; 0, 1, 1, 1; 0, 0, 0; 0; 0); 1), 312 (7842: 2 (4728), 18 (8429: 0, 0, 1, 0, 0; 1, 2 (4927), 4 (3487), 0; 1, 3 (9427), 4 (8127); 1; 1), 64 (3479: 1, 2 (4983), 10 (4761: 0, 0, 1, 0, 0; 1, 1, 1, 2 (4083); 1, 1, 1; 0; 1), 10 (4681), 3 (0128); 1, 3 (8493), 13 (4175: 1, 1, 2 (3781), 3 (8403), 0; 0, 0, 1, 1; 1, 1, 1; 0; 1), 7 (5481); 2 (3497), 3 (3498), 6 (3408: 0, 0, 0, 0, 0; 0, 0, 1, 2 (5471); 1, 0, 1; 0; 1); 2 (3489); 1), 68 (4195: 1, 4 (5401), 14 (9713: 1, 1, 2 (3409), 2 (0461), 0; 1, 1, 1, 1; 1, 1, 1; 0; 1), 8 (3081: 0, 1, 0, 0, 0; 1, 1, 0, 0; 1, 2 (3017), 0; 1; 1), 0; 3 (4519), 6 (4510), 13 (3791: 1, 1, 3 (9120), 1, 0; 1, 1, 1, 2 (4061); 0, 0, 1; 0; 1), 7 (0173: 0, 0, 0, 0, 0; 1, 2 (3108), 0, 0; 1, 1, 0; 1; 1); 2 (4591), 3 (4169), 5 (8193); 1; 1), 10 (9103: 3 (3091), 0, 0, 0, 0; 3 (0913), 0, 0, 0; 3 (9013), 0, 0; 0; 1); 4 (4782: 1, 0, 0, 0, 0; 2 (4827), 0, 0, 0; 0, 0, 0; 0; 1), 23 (8492: 1, 1, 3 (4873), 0, 0; 1, 3 (4082), 5 (4702), 0; 1, 3 (0482), 3 (8172); 1; 1), 50 (9712: 2 (7129), 2 (7021), 8 (7461: 1, 1, 1, 0, 0; 0, 0, 2 (7183), 1; 0, 0, 1; 0; 1), 11 (6148), 3 (4803); 0, 3 (0172), 10 (0492: 0, 0, 1, 2 (5741), 1; 1, 0, 1, 1; 1, 0, 1; 0; 1), 3 (3740); 1, 2 (8912), 3 (9743); 1; 1), 32 (5941: 0, 0, 6 (9012), 4 (0813), 0; 1, 2 (0145), 6 (7913), 1; 2 (9541), 3 (0541), 4 (0943); 2 (6941); 1); 3 (8742: 0, 0, 0, 0, 0; 2 (4872), 0, 0, 0; 0, 0, 0; 0; 1), 11 (7492: 0, 0, 1, 0, 0; 1, 2 (8942), 3 (8042), 0; 0, 1, 1; 1; 1), 20 (7129: 0, 1, 3 (0942), 3 (6841), 2 (0843); 1, 1, 4 (7641), 2 (3849); 1, 1, 1; 0; 0); 6 (2819*); 1), 84 (4178: 4 (7841: 1, 0, 0, 0, 0; 0, 0, 0, 0; 2 (7481), 0, 0; 0; 1), 20 (7419: 1, 3 (9841), 2 (0841), 0, 0; 1, 3 (9481), 2 (0481), 0; 2 (7491), 2 (0417), 1; 2 (7410); 1), 6 (9410: 1, 0, 0, 0, 0; 2 (0491), 0, 0, 0; 2 (0419), 0, 0; 0; 1), 0, 0; 5 (7418: 2 (8147), 0, 0, 0, 0; 2 (4817), 0, 0, 0; 0, 0, 0; 0; 1), 22 (4791: 2 (9147), 4 (0147), 2 (8140), 0, 0; 2 (9471), 3 (0471), 1, 0; 1, 2 (4981), 2 (4801); 2 (4891); 1), 6 (4019: 1, 0, 0, 0, 0; 2 (0149), 0, 0, 0; 2 (4910), 0, 0; 0; 1), 0; 4 (7148: 1, 0, 0, 0, 0; 2 (4718), 0, 0, 0; 0, 0, 0; 0; 1), 10 (4971: 0, 1, 1, 0, 0; 1, 2 (4189), 2 (4018), 0; 0, 1, 0; 1; 1), 2 (4109); 4 (4108); 1); 5 (6325: 2 (3652), 0, 0, 0, 0; 2 (5362), 0, 0, 0; 0, 0, 0; 0; 1), 65 (7328: 0, 8 (2683: 0, 1, 1, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (2583); 1), 8 (2603: 0, 1, 1, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (2503); 1), 14 (4652: 1, 3 (3546), 1, 0, 0; 1, 2 (6453), 1, 0; 1, 1, 1; 1; 1), 0; 0, 7 (8362: 1, 1, 0, 0, 0; 0, 0, 0, 0; 1, 2 (5382), 1; 0; 1), 8 (6392: 1, 1, 0, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (5392); 1), 7 (5361: 0, 0, 1, 0, 0; 1, 2 (5126), 1, 0; 0, 1, 0; 0; 1); 0, 4 (8325), 6 (0325: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (5329), 1; 1; 1); 3 (5328); 0), 201 (5371: 2 (7153), 13 (2718: 0, 1, 1, 6 (3150), 0; 0, 0, 1, 1; 0, 1, 1; 1; 0), 39 (2518: 1, 2 (9152), 6 (4752), 8 (3450), 1; 1, 2 (2105), 5 (2745), 4 (3016); 1, 2 (2915), 3 (2617); 2 (2519); 1), 40 (2548: 2 (4852), 4 (4952), 4 (4726), 4 (3496), 0; 1, 2 (2405), 6 (2983), 8 (2169: 0, 0, 1, 1, 0; 0, 1, 2 (2093), 0; 1, 0, 0; 1; 1); 1, 2 (2045), 3 (2168); 2 (2549); 1), 12 (8426: 1, 2 (2649), 0, 0, 0; 1, 2 (2460), 0, 0; 1, 2 (4026), 0; 2 (9426); 1); 2 (3751), 10 (8915: 0, 1, 1, 1, 0; 0, 1, 1, 2 (6317); 0, 1, 1; 1; 0), 33 (8327: 1, 2 (2073), 4 (2761), 2 (2501), 4 (6451); 1, 2 (7392), 2 (6318), 6 (4395); 1, 2 (7320), 3 (8345); 2 (0327); 1), 24 (8329: 0, 0, 4 (2961), 2 (2061), 0; 1, 3 (0382), 2 (6348), 2 (4360); 2 (9328), 3 (0328), 2 (6349); 2 (0329); 1); 1, 7 (2478: 0, 0, 1, 1, 2 (5310); 0, 0, 1, 1; 0, 0, 1; 0; 0), 13 (1368: 0, 0, 1, 0, 0; 1, 2 (6301), 1, 4 (0372); 1, 2 (9361), 1; 0; 0); 4 (5301: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 1; 2 (5391); 1); 1), 136 (2178: 0, 2 (7381), 22 (7391: 0, 0, 1, 2 (4816), 0; 1, 2 (0317), 5 (8310), 2 (4851); 1, 2 (9381), 3 (8301); 2 (7301); 1), 22 (4397: 0, 0, 2 (9416), 2 (0416), 0; 1, 2 (7340), 6 (9301: 0, 0, 1, 0, 0; 1, 0, 2 (8340), 0; 1, 0, 0; 0; 1), 2 (4051); 1, 2 (0347), 2 (0391); 1; 1), 4 (9340: 0, 0, 0, 0, 0; 1, 0, 0, 0; 2 (4390), 0, 0; 0; 1); 2 (2817), 16 (2791: 0, 0, 1, 0, 0; 1, 2 (2819), 4 (2487), 0; 1, 2 (2981), 2 (2081); 2 (2701); 1), 23 (2749: 0, 0, 2 (9371), 3 (0371), 1; 1, 1, 6 (2910), 1; 1, 2 (2047), 2 (2840); 2 (2740); 1), 16 (9146: 0, 0, 4 (2409), 2 (4370), 0; 0, 1, 3 (4150), 1; 1, 1, 1; 1; 1); 3 (2871: 0, 0, 0, 0, 0; 2 (2718), 0, 0, 0; 0, 0, 0; 0; 1), 9 (2189: 0, 0, 0, 0, 0; 1, 2 (2018), 2 (2071), 0; 0, 1, 1; 1; 1), 11 (2489: 0, 0, 2 (4158), 1, 0; 1, 1, 1, 0; 1, 1, 2 (2109); 1; 0); 5 (2170: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (2478); 1; 1); 1); 4 (2653: 1, 0, 0, 0, 0; 2 (2365), 0, 0, 0; 0, 0, 0; 0; 1), 28 (7328: 0, 2 (2853), 2 (2953), 5 (2516), 0; 0, 4 (8352), 6 (2360), 3 (5316); 0, 3 (8326), 2 (9326); 1; 0), 61 (7328: 0, 0, 12 (2157: 0, 0, 0, 0, 0; 0, 1, 3 (2476), 2 (2846); 1, 2 (2851), 1; 1; 1), 14 (2159: 0, 0, 0, 0, 0; 0, 1, 3 (2946), 2 (2046); 1, 2 (2051), 2 (2106); 2 (2150); 1), 1; 1, 6 (2370: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 1, 1; 2 (2380); 1), 10 (8346: 0, 0, 1, 1, 0; 0, 0, 1, 2 (2390); 1, 1, 1; 1; 1), 10 (9346: 0, 0, 0, 0, 0; 0, 1, 1, 1; 1, 1, 2 (0316); 2 (0346); 1); 1, 2 (2308), 4 (7346); 0; 0); 11 (3758: 0, 0, 2 (2376), 2 (2306), 0; 0, 1, 3 (2350), 2 (2156); 0, 1, 0; 0; 0); 1)

D = (2567: 6 (6752: 1, 0, 0, 0, 0; 2 (5672), 0, 0, 0; 2 (6725), 0, 0; 0; 1), 91 (8925: 0, 9 (7852: 0, 0, 0, 0, 0; 1, 2 (5792), 1, 0; 1, 1, 1; 1; 1), 13 (5602: 0, 0, 0, 0, 0; 1, 2 (0752), 3 (6872), 0; 1, 1, 3 (7682); 1; 1), 20 (5671: 1, 5 (7356), 1, 0, 0; 2 (5716), 2 (7653), 2 (7602), 0; 2 (7651), 2 (5476), 1; 1; 1), 0; 0, 10 (8652), 11 (6728: 1, 0, 0, 0, 0; 1, 2 (6972), 2 (5620), 0; 1, 1, 1; 1; 1), 13 (6745: 0, 0, 2 (7620), 0, 0; 1, 2 (7615), 1, 0; 2 (6475), 2 (6375), 1; 1; 1); 0, 5 (9725), 6 (7025); 4 (8725); 0), 369 (5641: 0, 24 (4856: 1, 3 (6495), 4 (6015), 0, 0; 1, 4 (6450), 2 (6159), 0; 1, 3 (0456), 2 (0156); 2 (4056); 1), 81 (8716: 1, 5 (6478: 0, 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 2 (6179); 0; 1), 14 (7159: 0, 0, 1, 2 (6385), 0; 1, 1, 2 (6479), 2 (6358); 1, 1, 1; 1; 1), 14 (7450: 1, 1, 1, 1, 0; 1, 1, 1, 2 (6953); 1, 1, 1; 1; 1), 0; 1, 8 (0176: 0, 0, 1, 0, 0; 0, 0, 1, 0; 1, 1, 2 (4876); 1; 1), 12 (4976: 0, 0, 0, 1, 0; 1, 1, 4 (4758), 1; 1, 1, 0; 1; 1), 8 (0356); 3 (6718), 6 (6710), 5 (0715); 3 (0716); 1), 75 (3875: 2 (8753), 8 (7359), 6 (6793), 8 (8926), 6 (0926: 1, 0, 0, 0, 0; 2 (6029), 0, 0, 0; 2 (6920), 0, 0; 0; 1); 3 (7385), 7 (7395: 1, 1, 1, 0, 0; 0, 0, 2 (8376), 0; 0, 0, 0; 1; 1), 12 (6073: 1, 1, 0, 1, 2 (8925); 1, 1, 0, 1; 1, 1, 0; 1; 1), 8 (9826); 2 (8375), 5 (3705), 4 (3976); 3 (3075); 1), 30 (7829: 3 (8972), 7 (8072: 0, 0, 0, 0, 0; 1, 2 (0792), 0, 0; 1, 1, 0; 1; 1), 0, 0, 0; 3 (9728), 7 (7082: 2 (0728), 1, 0, 0, 0; 1, 0, 0, 0; 0, 1, 0; 1; 1), 0, 0; 3 (8729), 4 (7028), 0; 2 (7820); 1); 0, 24 (8615: 2 (6851), 5 (5106), 2 (5406), 0, 0; 1, 4 (6845: 1, 0, 2 (5016), 0, 0; 0, 0, 0, 0; 0, 0, 0; 0; 1), 4 (6045), 0; 0, 1, 2 (4605); 2 (0615); 1), 54 (4678: 2 (7846), 9 (7946), 9 (0745: 0, 0, 1, 1, 0; 0, 0, 2 (8751), 1; 1, 1, 0; 1; 1), 8 (7051), 0; 1, 1, 9 (6071: 1, 1, 1, 2 (3685), 0; 0, 0, 1, 1; 0, 0, 0; 1; 1), 8 (3605); 0, 2 (7618), 2 (5178); 2 (4679); 1), 42 (5829: 0, 2 (8692), 4 (0692), 5 (3678), 3 (3670); 1, 4 (5902), 7 (5378), 4 (5703); 2 (5928), 4 (5028), 3 (0629); 2 (5029); 1); 0, 12 (8651: 0, 1, 2 (5046), 0, 0; 1, 2 (5619), 2 (0645), 0; 0, 1, 0; 2 (9651); 1), 21 (5718: 0, 2 (8671), 4 (7601), 2 (7640), 0; 1, 2 (5971), 2 (7648), 2 (5603); 1, 2 (5701), 2 (5740); 1; 0); 6 (1289: 0, 0, 1, 2 (5648), 1; 0, 0, 1, 1; 0, 0, 0; 0; 0); 0), 378 (5389: 4 (8953: 1, 0, 0, 0, 0; 0, 0, 0, 0; 2 (3958), 0, 0; 0; 1), 32 (8195: 1, 2 (9458), 8 (0853), 0, 0; 3 (8951), 1, 9 (3698: 0, 1, 1, 0, 0; 1, 1, 2 (3905), 0; 1, 0, 0; 1; 1), 0; 1, 2 (4895), 3 (8693); 1; 1), 94 (4058: 2 (8405), 5 (9405), 18 (3870: 1, 1, 1, 5 (0915), 0; 1, 1, 3 (9802), 1; 1, 1, 1; 0; 1), 20 (3690: 1, 3 (0973), 2 (8916), 2 (8971), 0; 2 (3906), 1, 2 (8196), 2 (7891); 1, 1, 1; 1; 1), 0; 3 (8450), 6 (9450), 16 (7948: 0, 1, 2 (8092), 2 (9150), 0; 1, 1, 1, 2 (3608); 2 (7498), 1, 1; 1; 1), 11 (6198: 0, 0, 0, 1, 0; 1, 2 (7918), 1, 1; 1, 1, 1; 1; 1); 2 (0458), 4 (0158), 6 (4698); 0; 1), 78 (6948: 0, 6 (4690), 16 (8601: 1, 2 (0196), 1, 3 (9470), 0; 1, 2 (9610), 1, 0; 2 (8610), 1, 0; 1; 1), 14 (8170: 1, 3 (7091), 0, 0, 0; 3 (7810), 2 (9071), 0, 0; 2 (8071), 1, 0; 1; 1), 0; 0, 8 (9046), 14 (7840: 3 (0478), 0, 1, 2 (0916), 0; 1, 1, 1, 1; 1, 1, 1; 0; 1), 7 (7910: 0, 1, 0, 0, 0; 1, 2 (0718), 0, 0; 1, 1, 0; 0; 1); 0, 4 (0946), 7 (6018); 2 (6048); 0), 0; 5 (3985: 2 (5893), 0, 0, 0, 0; 2 (8395), 0, 0, 0; 0, 0, 0; 0; 1), 32 (5918: 2 (8159), 3 (8459), 11 (8305: 0, 1, 3 (9783), 0, 0; 1, 1, 0, 0; 1, 1, 1; 1; 1), 0, 0; 1, 1, 9 (6398), 0; 1, 1, 1; 1; 1), 61 (3079: 1, 5 (6390), 11 (5901: 0, 0, 0, 1, 0; 1, 1, 2 (9781), 2 (6308); 1, 1, 1; 0; 1), 12 (9186: 0, 0, 2 (5801), 2 (5408), 0; 1, 0, 1, 1; 1, 1, 1; 1; 1), 0; 1, 4 (8370), 11 (7819: 0, 0, 2 (5091), 2 (3680), 0; 0, 1, 1, 1; 1, 0, 1; 1; 1), 7 (8649); 1, 1, 6 (8479); 0; 1), 30 (0179: 0, 1, 3 (4780), 3 (4086), 0; 1, 6 (7049), 5 (4609), 1; 2 (7109), 4 (0619), 2 (0186); 1; 1); 4 (8359: 1, 0, 0, 0, 0; 2 (9385), 0, 0, 0; 0, 0, 0; 0; 1), 12 (0359: 0, 1, 1, 0, 0; 1, 1, 5 (9386: 0, 0, 2 (5849), 0, 0; 1, 1, 0, 0; 0, 0, 0; 0; 1), 0; 0, 1, 1; 0; 1), 19 (1509: 0, 2 (5180), 1, 3 (7380), 0; 1, 1, 3 (6189), 4 (6489); 1, 1, 2 (7309); 0; 0); 6 (1320: 0, 0, 0, 1, 1; 0, 0, 1, 2 (7389); 0, 0, 1; 0; 0); 1), 54 (8190: 6 (9018: 1, 0, 0, 0, 0; 2 (9801), 0, 0, 0; 2 (0918), 0, 0; 0; 1), 16 (3809: 2 (9083), 2 (0948), 0, 0, 0; 2 (0389), 4 (4908), 0, 0; 3 (3089), 1, 0; 1; 1), 0, 0, 0; 6 (8901: 2 (9810), 0, 0, 0, 0; 3 (8019: 2 (9108), 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 0, 0; 0; 1), 14 (3980: 3 (0893), 4 (8049), 0, 0, 0; 2 (3098), 1, 0, 0; 1, 1, 0; 1; 1), 0, 0; 5 (8910: 1, 0, 0, 0, 0; 3 (9180), 0, 0, 0; 0, 0, 0; 0; 1), 4 (4890), 0; 2 (8490); 1); 6 (2675: 2 (7526), 0, 0, 0, 0; 3 (2756: 2 (5627), 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 0, 0; 0; 1), 83 (8529: 0, 12 (2758: 0, 1, 1, 0, 0; 1, 2 (2685), 1, 0; 1, 2 (2795), 1; 1; 1), 15 (2786: 1, 1, 1, 0, 0; 1, 1, 3 (2075), 0; 1, 1, 3 (2705); 1; 1), 19 (5763: 2 (3657), 3 (6457), 2 (2670), 0, 0; 1, 4 (7165), 2 (2706), 0; 1, 1, 1; 1; 1), 0; 0, 9 (6592: 1, 2 (5927), 1, 0, 0; 0, 0, 0, 0; 0, 1, 1; 2 (6582); 1), 9 (6927: 1, 1, 2 (7502), 0, 0; 0, 0, 1, 0; 1, 0, 1; 1; 1), 8 (6571: 0, 0, 1, 0, 0; 1, 1, 1, 0; 0, 2 (4576), 0; 1; 1); 0, 4 (8572), 4 (0526); 3 (6529); 0), 222 (8529: 3 (2958), 13 (2950: 0, 0, 1, 0, 0; 1, 2 (2805), 3 (2698), 0; 1, 1, 2 (2986); 1; 1), 47 (5947: 0, 1, 4 (9365), 8 (2870: 0, 0, 1, 0, 0; 1, 0, 1, 1; 2 (2078), 0, 1; 0; 1), 5 (2086: 1, 0, 0, 0, 0; 1, 0, 0, 0; 2 (2680), 0, 0; 0; 1); 1, 4 (9357), 4 (5468), 6 (5861); 2 (4957), 4 (5397), 5 (5187); 2 (5847); 1), 47 (7368: 1, 6 (6847), 8 (6197: 0, 0, 0, 0, 0; 1, 1, 0, 1; 1, 2 (6947), 0; 1; 1), 7 (5407: 0, 0, 0, 0, 0; 1, 1, 0, 0; 2 (5047), 1, 0; 1; 1), 0; 1, 2 (9763), 5 (5307), 6 (0165); 2 (3768), 3 (7963), 3 (0365); 2 (7168); 1), 18 (3607: 3 (7360), 5 (0761), 0, 0, 0; 1, 2 (6017), 0, 0; 1, 4 (0617), 0; 1; 1); 2 (2859), 9 (2789: 1, 1, 2 (9502), 0, 0; 0, 0, 1, 0; 1, 0, 1; 1; 1), 38 (3586: 1, 4 (8165: 0, 0, 0, 0, 0; 0, 0, 2 (5369), 0; 0, 0, 0; 1; 1), 5 (5169), 2 (2609), 4 (9027); 0, 3 (6548), 6 (7548), 2 (7591); 1, 3 (7583), 4 (9546); 2 (3596); 1), 25 (7510: 0, 0, 1, 4 (8647), 0; 1, 2 (0573), 6 (6503: 0, 0, 0, 2 (7169), 0; 0, 1, 0, 0; 1, 1, 0; 0; 1), 2 (7369); 1, 4 (4570), 1; 2 (6510); 1); 2 (9528), 4 (9520), 11 (4579: 0, 0, 0, 1, 0; 0, 0, 1, 1; 1, 2 (6549), 3 (8573); 1; 1); 2 (0529); 1), 126 (8369: 0, 4 (9583), 24 (9487: 0, 1, 3 (2890), 3 (3590), 0; 1, 1, 5 (3807), 3 (0583); 2 (9847), 1, 2 (9581); 1; 1), 26 (0487: 0, 1, 2 (9540), 2 (9510), 0; 1, 5 (4907), 3 (0518), 1; 2 (4087), 3 (9407), 3 (0917); 2 (0187); 1), 0; 2 (3968), 13 (3960: 0, 0, 2 (8593), 0, 0; 1, 1, 4 (9861), 0; 1, 1, 1; 1; 1), 23 (0961: 0, 0, 5 (2809), 3 (8497), 0; 1, 1, 5 (8591: 0, 0, 0, 2 (4068), 0; 0, 1, 0, 1; 0, 0, 0; 0; 1), 2 (8947); 1, 2 (4960), 1; 1; 1), 10 (8107: 0, 0, 1, 1, 0; 0, 1, 1, 1; 1, 2 (8047), 0; 1; 1); 3 (8963: 0, 0, 0, 0, 0; 2 (3869), 0, 0, 0; 0, 0, 0; 0; 1), 7 (3069), 9 (4069: 0, 0, 0, 1, 0; 0, 1, 2 (8160), 1; 1, 1, 1; 0; 1); 4 (0369: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (8469); 0; 1); 1); 5 (2657: 1, 0, 0, 0, 0; 3 (2765), 0, 0, 0; 0, 0, 0; 0; 1), 31 (8529: 0, 4 (2865), 5 (2697), 5 (1320*), 0; 0, 4 (2586), 4 (0562), 4 (6517); 0, 3 (2579), 1; 1; 0), 57 (8269: 0, 3 (2598), 7 (2907), 6 (9547: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (3597), 1; 1; 1), 4 (4507); 1, 4 (2860), 12 (3967: 0, 0, 0, 1, 0; 0, 1, 1, 1; 1, 2 (9167), 2 (3568); 2 (4967); 1), 11 (0367: 0, 0, 0, 0, 0; 0, 1, 1, 2 (8517); 1, 2 (4067), 1; 2 (0167); 1); 1, 1, 7 (8367: 0, 0, 0, 0, 0; 0, 0, 1, 1; 0, 1, 1; 2 (8467); 1); 0; 0); 11 (3268: 0, 0, 1, 2 (2507), 0; 0, 1, 4 (2569), 1; 0, 1, 1; 0; 0); 1)

E = (5678: 9 (6785: 2 (7856), 0, 0, 0, 0; 4 (6857: 1, 0, 0, 0, 0; 2 (7865), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 2 (6587), 0, 0; 0; 1), 88 (6789: 3 (8967), 17 (8965: 1, 2 (9857), 1, 0, 0; 1, 4 (9567), 2 (0867), 0; 2 (8956), 1, 1; 1; 1), 16 (7056: 1, 3 (0865), 0, 0, 0; 1, 4 (8065), 0, 0; 2 (7506), 3 (7850), 0; 1; 1), 0, 0; 4 (7869: 1, 0, 0, 0, 0; 2 (7986), 0, 0, 0; 0, 0, 0; 0; 1), 16 (9586: 1, 3 (8795), 2 (8760), 0, 0; 1, 4 (9765), 1, 0; 0, 1, 1; 1; 1), 12 (0587: 2 (8750), 2 (6805), 0, 0, 0; 1, 3 (0756), 0, 0; 1, 1, 0; 1; 1), 0; 3 (8769), 8 (8759: 0, 1, 1, 0, 0; 2 (9785), 1, 1, 0; 0, 1, 0; 0; 1), 5 (6085: 0, 0, 0, 0, 0; 0, 1, 0, 0; 1, 2 (6705), 0; 0; 1); 3 (6589); 1), 84 (6790: 4 (0967: 0, 0, 0, 0, 0; 2 (7069), 0, 0, 0; 1, 0, 0; 0; 1), 24 (7059: 1, 5 (9506), 3 (9806), 0, 0; 2 (0957), 4 (0956), 2 (9086), 0; 2 (7509), 2 (7809), 1; 1; 1), 8 (8509: 2 (9085), 0, 0, 0, 0; 2 (0859), 0, 0, 0; 3 (8905), 0, 0; 0; 1), 0, 0; 5 (0769: 2 (7096), 0, 0, 0, 0; 2 (9706), 0, 0, 0; 0, 0, 0; 0; 1), 21 (8096: 2 (6809), 5 (6509), 3 (9705), 0, 0; 2 (6089), 3 (0897), 1, 0; 1, 1, 1; 1; 1), 5 (9850: 1, 0, 0, 0, 0; 1, 0, 0, 0; 2 (8950), 0, 0; 0; 1), 0; 4 (9760: 1, 0, 0, 0, 0; 2 (0796), 0, 0, 0; 0, 0, 0; 0; 1), 8 (6950: 0, 1, 0, 0, 0; 1, 1, 2 (9780), 0; 0, 1, 0; 1; 1), 1; 3 (8790); 1), 0, 0; 8 (1567: 0, 4 (6875: 1, 0, 0, 0, 0; 2 (7685), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 2 (8657), 0, 0; 0, 2 (5867), 0; 0; 0), 72 (5689: 3 (9568), 15 (7968: 1, 2 (9875), 0, 0, 0; 2 (8976), 3 (9758), 2 (6508), 0; 1, 1, 1; 1; 1), 15 (6870: 2 (7068), 3 (0758), 0, 0, 0; 2 (8076), 2 (8075), 0, 0; 1, 3 (8570), 0; 1; 1), 0, 0; 2 (8695), 13 (5796: 2 (6579), 2 (8579), 2 (8605), 0, 0; 2 (7695), 1, 1, 0; 0, 1, 1; 0; 1), 8 (5760: 2 (7605), 1, 0, 0, 0; 1, 1, 0, 0; 2 (5067), 0, 0; 0; 1), 0; 4 (9685: 1, 0, 0, 0, 0; 2 (8659), 0, 0, 0; 0, 0, 0; 0; 1), 6 (9687), 4 (5087); 2 (7689); 0), 48 (5790: 3 (9075), 13 (6079: 1, 2 (7908), 2 (9508), 0, 0; 2 (0976), 1, 1, 0; 1, 1, 0; 1; 1), 5 (6908: 1, 0, 0, 0, 0; 2 (9068), 0, 0, 0; 1, 0, 0; 0; 1), 0, 0; 2 (5907), 13 (9870: 1, 3 (0598), 3 (5069), 0, 0; 1, 1, 0, 0; 1, 1, 1; 0; 1), 2 (6098), 0; 2 (5709), 5 (0798), 1; 1; 1), 0; 6 (5687: 1, 0, 0, 0, 0; 4 (5768: 1, 0, 0, 0, 0; 2 (5876), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 0, 0; 0; 1), 24 (5689: 0, 2 (9578), 2 (6078), 0, 0; 2 (9658), 6 (7698), 6 (5076: 0, 1, 0, 0, 0; 1, 2 (5708), 0, 0; 0, 1, 0; 0; 1), 0; 0, 3 (8679), 1; 1; 1), 12 (5790: 0, 3 (0978), 1, 0, 0; 1, 3 (5908), 1, 0; 1, 1, 0; 1; 0); 8 (1579: 0, 0, 1, 1, 0; 0, 1, 3 (5078), 1; 0, 1, 0; 0; 0); 1)

F = (1245: 0, 4 (4132: 1, 0, 0, 0, 0; 2 (3124), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 2 (4213), 0, 0; 0, 2 (3241), 0; 0; 0)

G = (1356: 0, 17 (4135: 1, 2 (5213), 2 (6213), 0, 0; 1, 4 (3215), 1, 0; 1, 2 (4631), 1; 1; 1), 64 (3178: 0, 8 (7431: 0, 0, 1, 0, 0; 0, 1, 0, 0; 1, 2 (2731), 1; 1; 1), 8 (0431: 0, 0, 1, 0, 0; 0, 1, 0, 0; 1, 2 (4931), 1; 1; 1), 14 (4263: 1, 3 (2614), 1, 0, 0; 1, 2 (6241), 1, 0; 1, 1, 1; 1; 1), 0; 0, 8 (4137), 12 (3219: 1, 2 (0132), 1, 0, 0; 1, 1, 0, 0; 1, 2 (3914), 1; 1; 1), 6 (3524); 0, 5 (2138), 2 (3104); 1; 0), 52 (7283: 0, 7 (2437), 12 (2439: 1, 1, 1, 0, 0; 1, 1, 3 (2174), 0; 1, 1, 0; 1; 1), 6 (0124), 0; 0, 3 (7432), 8 (8241: 1, 1, 0, 0, 0; 1, 1, 0, 0; 0, 1, 2 (3240); 0; 1), 6 (9241); 0, 2 (4273), 6 (0243); 2 (7243); 0), 0; 0, 12 (1263: 1, 2 (6314), 2 (5314), 0, 0; 1, 1, 0, 0; 1, 2 (1643), 1; 1; 0), 29 (7483: 0, 1, 6 (2364: 1, 0, 0, 0, 0; 0, 0, 0, 0; 1, 1, 2 (9314); 0; 1), 5 (4251), 0; 0, 3 (1743), 6 (1723: 0, 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 2 (1043); 1; 1), 4 (1923); 0, 1, 2 (1493); 1; 0), 20 (7328: 0, 0, 4 (1842), 4 (1042), 0; 0, 2 (2374), 3 (2304), 2 (1420); 0, 1, 3 (9324); 1; 0); 0, 4 (1453), 15 (3728: 0, 2 (1372), 4 (1302), 3 (1340), 0; 0, 1, 3 (1320), 1; 0, 1, 0; 0; 0); 3 (1326); 0)

H = (1356: 3 (6135), 41 (7581: 0, 3 (8135), 9 (5139: 0, 0, 1, 0, 0; 0, 0, 1, 0; 1, 1, 3 (6138); 1; 1), 8 (4635: 0, 1, 0, 0, 0; 0, 1, 2 (6139), 0; 1, 1, 0; 1; 1), 0; 0, 3 (5831), 7 (8631: 0, 0, 1, 0, 0; 0, 0, 1, 0; 1, 1, 2 (5931); 0; 1), 6 (9631); 0, 1, 3 (9531); 1; 0), 165 (7532: 2 (3275), 10 (3895: 0, 1, 1, 3 (5217), 0; 0, 1, 0, 1; 0, 1, 1; 1; 0), 27 (6283: 0, 0, 3 (5624), 0, 2 (5174); 1, 2 (3260), 3 (4265), 6 (5291); 1, 2 (0263), 4 (6217); 2 (6203); 1), 23 (6218: 0, 0, 3 (5184), 6 (3064), 0; 1, 2 (0261), 2 (6174), 2 (5914); 1, 2 (6291), 1; 2 (6210); 1), 12 (6184: 0, 0, 0, 0, 0; 1, 2 (9614), 0, 0; 2 (8164), 4 (0164), 0; 2 (6194); 1); 1, 9 (8437: 0, 1, 1, 1, 0; 0, 1, 1, 2 (2935); 0, 0, 1; 1; 0), 26 (4835: 1, 2 (3594), 2 (2564), 0, 1; 1, 2 (5430), 4 (8137), 6 (0731); 1, 2 (0435), 1; 2 (4935); 1), 23 (8139: 0, 0, 1, 2 (7614), 0; 1, 3 (0831), 3 (4638), 3 (6430); 2 (9138), 3 (9130), 2 (6439); 2 (8130); 1); 2 (2537), 9 (2538: 0, 0, 0, 0, 0; 1, 2 (5932), 2 (7435), 0; 0, 0, 1; 2 (2539); 1), 16 (6832: 0, 0, 0, 0, 1; 0, 0, 2 (4538), 6 (7931); 1, 2 (9632), 1; 2 (6032); 1); 4 (8532: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 1; 2 (0532); 1); 1), 180 (7283: 3 (2837), 18 (2937: 0, 0, 1, 0, 0; 1, 2 (9832), 3 (0832), 0; 1, 3 (8932), 4 (2830); 2 (2037); 1), 36 (9437: 2 (3794), 3 (3704), 8 (2574), 5 (2864), 0; 1, 3 (4938), 5 (2039), 0; 1, 2 (4037), 3 (0438); 2 (0437); 1), 30 (6924: 0, 0, 3 (9430), 0, 0; 1, 3 (2064), 7 (8194), 5 (0714); 2 (9624), 2 (0624), 4 (5024); 2 (6024); 1), 4 (0914: 0, 0, 0, 0, 0; 1, 0, 0, 0; 2 (0194), 0, 0; 0; 1); 2 (7832), 12 (3279: 1, 1, 1, 0, 0; 0, 0, 2 (3784), 0; 1, 2 (3298), 2 (8271); 1; 1), 29 (9218: 0, 0, 3 (2584), 4 (7524), 1; 1, 1, 5 (3209), 4 (5247); 1, 2 (9271), 4 (6248); 2 (0218); 1), 20 (4295: 0, 0, 2 (9184), 3 (0184), 0; 1, 2 (6249), 5 (9201), 0; 1, 2 (4269), 2 (0291); 1; 1); 2 (3287), 7 (8203), 11 (7219: 0, 0, 0, 0, 0; 0, 1, 4 (9203), 1; 1, 1, 1; 1; 1); 5 (7203: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (7281); 1; 1); 1), 48 (2784: 3 (8247), 14 (4279: 0, 0, 0, 0, 0; 1, 3 (0247), 2 (8240), 0; 2 (4297), 3 (8249), 1; 1; 1), 4 (4209: 0, 0, 0, 0, 0; 1, 0, 0, 0; 2 (0249), 0, 0; 0; 1), 0, 0; 2 (4287), 8 (9824: 1, 1, 0, 0, 0; 0, 0, 0, 0; 1, 2 (7924), 1; 1; 1), 2 (0924), 0; 2 (8724), 6 (2894), 2 (2094); 4 (2704); 1); 2 (5136), 29 (7683: 0, 1, 6 (2536: 0, 1, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (0136); 1; 1), 5 (6251), 0; 0, 3 (1763), 5 (1063), 4 (1503: 0, 0, 0, 0, 0; 0, 0, 2 (1625), 0; 0, 0, 0; 1; 1); 0, 1, 3 (1693); 1; 0), 110 (1678: 0, 2 (8216), 16 (8436: 1, 1, 0, 1, 1; 1, 1, 1, 3 (9216); 1, 2 (2836), 1; 1; 1), 23 (9436: 1, 2 (0364), 3 (3754), 4 (8253), 1; 1, 1, 3 (5246), 1; 1, 2 (2936), 1; 1; 1), 8 (0253); 0, 4 (1862), 20 (1745: 0, 0, 1, 1, 0; 0, 1, 3 (1460), 6 (1803); 1, 1, 3 (1793); 2 (1725); 1), 17 (1495: 0, 0, 2 (2654), 1, 0; 1, 2 (1529), 3 (1520), 0; 1, 3 (1592), 2 (1093); 1; 1); 0, 5 (1682), 12 (1629: 0, 0, 0, 0, 0; 0, 0, 2 (1572), 3 (1073); 1, 1, 2 (1640); 2 (1620); 1); 3 (1628); 0), 76 (1728: 0, 0, 8 (4286), 16 (4259: 0, 0, 4 (8394), 4 (8304), 0; 1, 2 (2054), 1, 0; 0, 1, 1; 1; 1), 2 (0394); 1, 8 (1892: 0, 0, 0, 0, 0; 0, 0, 1, 0; 1, 2 (1082), 2 (1847); 1; 1), 14 (1479: 0, 0, 1, 1, 0; 1, 1, 3 (1902), 0; 1, 2 (1849), 1; 2 (1489); 1), 4 (1940: 0, 0, 0, 0, 0; 1, 0, 0, 0; 2 (1490), 0, 0; 0; 1); 2 (1782), 7 (1027), 8 (8749*); 5 (1720: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (1928); 1; 1); 1); 2 (1365), 17 (5378: 0, 2 (1853), 3 (1953), 3 (1526), 0; 0, 1, 4 (1305), 2 (1369); 0, 1, 1; 0; 0), 37 (1738: 0, 0, 2 (8354), 2 (0354), 1; 1, 6 (1397: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (1370), 1; 1; 1), 8 (1846: 0, 0, 0, 0, 0; 0, 0, 1, 2 (1390); 1, 1, 1; 1; 1), 10 (1946: 0, 0, 0, 0, 0; 0, 1, 1, 1; 1, 1, 2 (1952); 2 (1046); 1); 1, 2 (1308), 4 (1458); 0; 0); 9 (3758: 0, 0, 2 (1376), 2 (1396), 0; 0, 1, 2 (1359), 1; 0, 1, 0; 0; 0); 1)

I = (1567: 0, 9 (5674: 0, 2 (6735), 0, 0, 0; 1, 3 (7635), 0, 0; 1, 1, 0; 0; 1), 99 (8256: 0, 3 (5684), 17 (6935: 0, 1, 1, 3 (7684), 0; 1, 1, 2 (5738), 2 (6784); 1, 1, 2 (6738); 1; 1), 22 (6794: 0, 1, 2 (7630), 1, 0; 1, 3 (5974), 3 (5739), 2 (0735); 2 (6974), 2 (6739), 2 (6730); 2 (6704); 1), 0; 1, 8 (2795: 0, 1, 1, 2 (6854), 0; 0, 1, 1, 1; 0, 1, 0; 0; 0), 18 (5279: 0, 0, 4 (6954), 3 (0654), 0; 1, 1, 1, 2 (8674); 1, 1, 1; 2 (5270); 1), 8 (9736); 2 (6258), 8 (5296: 0, 0, 1, 0, 0; 1, 1, 2 (7258), 0; 0, 0, 1; 1; 1), 8 (7296: 0, 0, 0, 1, 0; 0, 0, 1, 0; 1, 2 (0276), 1; 1; 1); 3 (0256); 1), 126 (8259: 0, 4 (9835), 22 (6894: 0, 1, 2 (7938), 4 (9035), 0; 1, 4 (6938), 2 (5904), 2 (0835); 1, 1, 2 (5804); 1; 1), 30 (0874: 0, 1, 3 (9730), 3 (9036), 0; 1, 6 (7094), 5 (6094), 1; 2 (0784), 4 (0794), 2 (0836); 1; 1), 0; 2 (5298), 13 (5290: 0, 0, 2 (9854), 0, 0; 1, 1, 3 (9286), 0; 1, 1, 2 (6298); 1; 1), 22 (6290: 0, 0, 4 (8936), 4 (8035), 0; 1, 1, 4 (6839: 0, 1, 0, 2 (0278), 0; 0, 0, 0, 0; 0, 0, 0; 0; 1), 1; 1, 2 (6208), 1; 2 (7290); 1), 10 (6039: 0, 0, 1, 1, 0; 0, 1, 1, 1; 1, 2 (8036), 0; 1; 1); 3 (8295: 0, 0, 0, 0, 0; 2 (9258), 0, 0, 0; 0, 0, 0; 0; 1), 8 (5209: 0, 0, 1, 0, 0; 1, 1, 1, 0; 0, 1, 2 (6289); 0; 1), 8 (7209); 3 (0259); 1), 18 (8249: 0, 4 (9804: 0, 0, 0, 0, 0; 1, 0, 0, 0; 2 (9084), 0, 0; 0; 1), 3 (0938), 0, 0; 0, 5 (0298), 2 (0839), 0; 0, 2 (8290), 1; 1; 0); 2 (1675), 33 (1678: 0, 0, 4 (5764: 0, 2 (6257), 0, 0, 0; 0, 1, 0, 0; 0, 0, 0; 0; 1), 0, 0; 1, 4 (1856: 0, 0, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2 (1706); 0; 1), 8 (1056: 0, 0, 2 (5637), 0, 0; 0, 1, 1, 0; 0, 1, 1; 1; 1), 0; 1, 5 (1976: 0, 0, 0, 0, 0; 0, 0, 2 (1685), 0; 0, 0, 1; 1; 1), 6 (1659); 3 (1670); 1), 90 (1589: 0, 0, 10 (5864: 0, 0, 1, 1, 0; 0, 1, 1, 2 (5297); 0, 1, 1; 1; 1), 16 (8637: 0, 2 (7864), 2 (9764), 3 (5064), 0; 0, 1, 1, 2 (5207); 1, 1, 1; 1; 1), 6 (7064: 0, 2 (0637), 0, 0, 0; 0, 1, 0, 0; 1, 1, 0; 0; 1); 2 (1895), 10 (1805: 0, 0, 0, 0, 0; 1, 1, 3 (1698), 0; 1, 1, 1; 1; 1), 20 (1690: 0, 0, 2 (9536), 3 (8536), 3 (7538); 1, 1, 3 (1708), 1; 1, 2 (1970), 1; 1; 1), 8 (6504: 0, 0, 0, 1, 0; 0, 1, 1, 1; 0, 2 (0574), 0; 1; 1); 2 (1985), 4 (1986), 10 (1079: 0, 0, 0, 1, 1; 0, 1, 2 (7539), 1; 1, 1, 1; 0; 1); 2 (1689); 0), 42 (8279: 0, 1, 10 (1986: 0, 1, 1, 2 (9037), 0; 0, 2 (1890), 1, 1; 0, 1, 0; 1; 0), 8 (0964), 0; 1, 5 (9207), 7 (1094*), 4 (8064); 1, 1, 3 (0269); 1; 0); 3 (1657: 0, 0, 0, 0, 0; 2 (1576), 0, 0, 0; 0, 0, 0; 0; 1), 21 (1578: 0, 0, 1, 0, 0; 1, 4 (1957), 8 (1760), 0; 0, 2 (1586), 2 (1596); 2 (1570); 1), 27 (8169: 0, 3 (1987), 7 (1907), 1, 1; 1, 4 (1068), 3 (9267), 3 (0267); 1, 1, 2 (8267); 0; 0); 9 (5869: 0, 0, 2 (1597), 1, 0; 0, 1, 2 (1967), 1; 0, 1, 1; 0; 0); 1)

J = (1243: 1, 0, 0, 0, 0; 4 (4231: 1, 0, 0, 0, 0; 2 (1432), 0, 0, 0; 0, 0, 0; 0; 1), 0, 0, 0; 0, 0, 0; 0; 1)

K = (1256: 0, 0, 4 (2634), 8 (2718: 0, 0, 2 (8134), 2 (0134), 0; 0, 0, 1, 2 (2934); 0, 0, 1; 0; 0), 0; 0, 8 (5214), 20 (1728: 0, 4 (8231), 4 (0231), 2 (3264), 0; 0, 1, 2 (1932), 2 (1435); 0, 2 (1824), 2 (1924); 1; 0), 16 (1738: 0, 0, 2 (3274), 2 (3204), 0; 0, 2 (1374), 3 (1304), 2 (4239); 0, 1, 3 (1439); 1; 0); 0, 2 (1245), 12 (1378: 0, 0, 0, 2 (3254), 0; 0, 1, 4 (1293), 2 (1249); 0, 1, 2 (1248); 0; 0); 2 (1246); 0)

L = (1356: 0, 3 (6534), 29 (5748: 0, 2 (8534), 7 (6834: 0, 0, 0, 0, 0; 0, 0, 1, 1; 1, 1, 2 (9534); 0; 1), 7 (6934), 2 (6239); 0, 1, 5 (2739*), 3 (5239); 0, 1, 1; 0; 0), 36 (7283: 0, 1, 6 (8034: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (9834), 1; 1; 1), 2 (0934), 0; 1, 7 (9238: 0, 0, 0, 0, 0; 0, 0, 1, 0; 1, 1, 1; 2 (0238); 1), 7 (3269*), 6 (0264: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (6294), 1; 1; 1); 1, 2 (7239), 3 (7264); 0; 0), 12 (7284: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 4 (8204), 2 (9204); 4 (9284); 1); 1, 15 (3578: 0, 2 (1835), 4 (1035), 3 (1639), 0; 0, 1, 3 (1539), 1; 0, 1, 0; 0; 0), 37 (1378: 0, 0, 2 (8236), 2 (0236), 1; 1, 6 (1037: 0, 0, 0, 0, 0; 0, 0, 0, 0; 1, 2 (1830), 1; 1; 1), 8 (1864: 0, 0, 0, 0, 0; 0, 0, 1, 2 (1930); 1, 1, 1; 1; 1), 10 (1694: 0, 0, 0, 0, 0; 0, 1, 1, 1; 1, 1, 2 (1295); 2 (1604); 1); 1, 2 (1038), 4 (1268); 0; 0), 28 (1478: 0, 0, 2 (7254), 2 (9254), 0; 1, 7 (1894: 0, 0, 0, 0, 0; 0, 0, 1, 0; 1, 1, 1; 2 (1804); 1), 6 (1207: 0, 0, 0, 0, 0; 0, 0, 1, 0; 0, 1, 2 (1904); 1; 1), 2 (1290); 1, 2 (1074), 4 (1279); 1; 0); 1, 5 (1738: 0, 0, 0, 0, 0; 0, 0, 0, 1; 0, 1, 2 (1936); 1; 0), 12 (2758: 0, 0, 2 (1286), 2 (1296), 0; 0, 1, 3 (1259), 2 (1054); 0, 1, 1; 0; 0); 1; 0)

M = (1356: 0, 0, 2 (6234), 4 (7128: 0, 0, 1, 2 (9234), 0; 0, 0, 1, 0; 0, 0, 0; 0; 0), 0; 0, 3 (1235), 9 (2738: 0, 0, 0, 1, 0; 0, 1, 3 (1230), 2 (1934); 0, 1, 1; 0; 0), 4 (1278: 0, 0, 0, 0, 0; 0, 0, 0, 0; 0, 1, 2 (1204); 1; 0); 0, 1, 1; 0; 0)
}